Adjacent vertex-distinguishing edge colorings of -minor free graphs

The adjacent vertex-distinguishing chromatic index χ a ′ ( G ) of a graph G is the smallest integer k for which G admits a proper edge k -coloring such that any pair of adjacent vertices are incident to distinct sets of colors. s paper, we prove that if G is a K 4 -minor free graph, without isolated edges, and with maximum degree Δ ≥ 4 , then Δ ≤ χ a ′ ( G ) ≤ Δ + 1 ; when Δ ≥ 5 , χ a ′ ( G ) = Δ if and only if G contains no adjacent vertices of maximum degree.